Improving measurements of the falling trajectory and terminal velocity of wind‐dispersed seeds

Abstract Seed dispersal by wind is one of the most important dispersal mechanisms in plants. The key seed trait affecting seed dispersal by wind is the effective terminal velocity (hereafter “terminal velocity”, V t ), the maximum falling speed of a seed in still air. Accurate estimates of V t are crucial for predicting intra‐ and interspecific variation in seed dispersal ability. However, existing methods produce biased estimates of V t for slow‐ or fast‐falling seeds, fragile seeds, and seeds with complex falling trajectories. We present a new video‐based method that estimates the falling trajectory and V t of wind‐dispersed seeds. The design involves a mirror that enables a camera to simultaneously record a falling seed from two perspectives. Automated image analysis then determines three‐dimensional seed trajectories at high temporal resolution. To these trajectories, we fit a physical model of free fall with air resistance to estimate V t . We validated this method by comparing the estimated V t of spheres of different diameters and materials to theoretical expectations and by comparing the estimated V t of seeds to measurements in a vertical wind tunnel. V t estimates closely match theoretical expectations for spheres and vertical wind tunnel measurements for seeds. However, our V t estimates for fast‐falling seeds are markedly higher than those in an existing trait database. This discrepancy seems to arise because previous estimates inadequately accounted for seed acceleration. The presented method yields accurate, efficient, and affordable estimates of the three‐dimensional falling trajectory and terminal velocity for a wide range of seed types. The method should thus advance the understanding and prediction of wind‐driven seed dispersal.


| INTRODUC TI ON
Seed dispersal by wind is one of the most important and best understood dispersal mechanisms (Nathan et al., 2011). A seed (or diaspore) released from the mother plant starts falling to the ground and accelerates until it reaches a terminal falling velocity (V t ) at which the opposing forces of gravitation and aerodynamic drag cancel out. V t is the most important seed trait determining wind-driven seed dispersal (Nathan et al., 2011). This is because seeds with lower V t are transported further by horizontal wind components and are more likely to be uplifted by turbulent airflows that can disperse seeds over long distances (Higgins et al., 2003;Nathan et al., 2002Nathan et al., , 2011. A method to measure V t should ideally (1) accurately measure V t of single seeds, (2) work for various seed types, (3) estimate V t of seeds that still accelerate, and (4) be nondestructive so that seeds can be used for further measurements. To our knowledge, no existing method fulfills all these criteria.
Existing methods for measuring V t fall into three categories.
The first lets seeds accelerate over a certain distance and then estimates V t as vertical seed velocity in a measuring section (Sullivan et al., 2018). This approach is nondestructive, but yields biased V t estimates for seeds that still accelerate. A second method measures the air speed needed to suspend seeds in vertical air flow (Jongejans & Schippers, 1999). This avoids bias from seed acceleration. While the method can accurately measure the mean V t of seed bulks, it is challenging to apply to single or slow-falling seeds because it is difficult to accurately control and measure vertical wind speed (Russo, 2011). Vertical wind tunnels may also damage fragile seeds and are challenging to use for seeds with complex falling trajectories (such as auto-rotating seeds). A third method includes video-based approaches that work for different seed types (Loubet et al., 2007), and are nondestructive (Wyse et al., 2019).
Existing video-based methods have two main drawbacks: first, they cannot accurately determine a seed's vertical displacement because they only film the seed from a single perspective.
Consequently, a given pixel displacement between two video frames may correspond to different seed displacements, depending on the seed-camera distance (Figure 1a). To account for this, Wyse et al. (2019) estimated the true vertical displacement by recording the landing location of seeds. However, this is labor-intensive and may still bias V t estimates when seeds show substantial lateral movement. Secondly, existing video-based methods fail to account for seed acceleration (Liu et al., 2021) because they estimate V t as the average velocity in the video (Gómez-Noguez et al., 2017). This may substantially underestimate V t for fast-falling seeds that take longer to approach V t .
Here, we present a new video-based method that accounts for acceleration and enables nondestructive estimation of falling trajectories and V t for a broad range of seed types. We demonstrate how to determine the intraspecific variability and repeatability of V t estimates. We then compare the V t estimates with measurements of V t in an existing database.

| DE S I G N OF THE APPAR ATUS
The apparatus consists of a seed release device and a box ( Figure S1).
The seed release device is a polyvinyl chloride tube (diameter: 10 cm) of adjustable length with an electromagnetic opening flap ( Figure S1).
When the flap opens, a seed is released and falls through the tube into the box. The box (1.2 × 0.84 × 0.5 m, width × depth × height) contains the falling corridor of 0.25 × 0.25 × 0.33 m (width × depth × height), which is bounded on two adjacent sides by two semitransparent plastic boards that are backlit by two battery-driven LED lamps to optimize the visual detectability of seeds (Figure 1a,c). To avoid air turbulence due to possible heating, we separated the lamps from the falling corridor with the plastic boards and ensured that the lamps are well ventilated. On the third side of the falling corridor is a highspeed monochrome camera with a rate of up to 150 frames per second (fps) (acA1920-155um, Basler AG), 58 cm from the lens to the center of the falling corridor ( Figure 1). On the fourth side, a mirror mounted at 62° from the camera plane enables the camera to detect the seed from two perspectives (  Videos of falling seeds are analyzed automatically with ImageJ (Schneider et al., 2012). The analysis script (Appendix S1) comprises four steps. First, a stack of inverted images is extracted from each video. Secondly, the first seed-free image of the stack is subtracted from every image to obtain differential images in which seeds stand out as dark shapes. Thirdly, the differential images are converted to binary images. Fourthly, for all objects above a threshold size, the image coordinates of the object center, as well as the object's area and circularity, are calculated. R function calculate.terminal.velocity.phys (in package velocimeter) then cleans the ImageJ output in four steps. First, it removes objects at the very edges of the falling corridor. Secondly, the largest object in both the direct and the mirror parts of the image is selected. Thirdly, the function considers the resulting pair of objects in the direct and mirror parts as representing a putative seed if their vertical coordinates (z d and z m ) are sufficiently close. Fourthly, the function selects the longest sequence of consecutive images containing a putative seed as depicting the seed trajectory. After data cleaning, the function uses the abovementioned conversion functions to derive a time series of three-dimensional seed coordinates.

| Estimating terminal velocity
Estimation of V t requires fitting a model to the measured seed trajectory. This can be an implicit model (such as when estimating V t as the velocity at the bottom of the falling corridor; Askew et al., 1997), a phenomenological asymptotic model, a simple or a complex physical model. In the following, we use a simple physical model of vertical F I G U R E 1 Video-based measurement of three-dimensional trajectories and terminal velocities (V t ) of falling seeds. (a) Video-based measurement faces the challenge that two seeds at different horizontal distances from the camera (A and B) may show the same pixel displacement between video frames even though they differ in true seed displacement (d A and d B ) and hence in falling velocity. (b) The presented apparatus circumvents this problem by using a mirror (light blue) that enables the camera to simultaneously record a seed from two perspectives. (c,d) Even though seeds A and B have identical positions in the direct image, their positions in the mirror image differ (A' and B′). (e) A video frame showing a seed of Ailanthus altissima in the direct and mirror image (blue and red, respectively). The seed center is represented by two coordinates in the direct image (x d and z d ) as well as two coordinates in the mirror image (y m and z m ). From these four image coordinates, the presented algorithm reconstructs the seed's three-dimensional position and estimates the effective V t .
x y free fall with air resistance, which assumes that a seed is a sphere (Taylor, 2005) (for details, see Appendix S2). This model assumes that the falling object is a sphere with a Reynolds number Re > 10 (so that drag is a quadratic function of velocity). We note, however, that alternative models can be used to estimate V t . The model predicts the vertical distance traveled over time t as where z(0) is position at t = 0, and g is the gravitational acceleration (9.81 m/s 2 ). We estimated V t and z (0)  We used the evaluation functions to assess performance of the simple physical model for seeds of five species with very different morphology (Table S2, Figure S4; n = 40 seeds per species).
The physical model fitted individual falling trajectories very well (R 2 > .996 for all seeds; Figure S5a), and the median RMSE falling velocity per species was <0.06 m/s, indicating a good to very good approximation of falling velocities ( Figure S6).
An introductory video shows how to use the apparatus, analyze the obtained videos, and estimate V t (Zhu et al., 2021).

| VALIDATION E XPERIMENTS
To validate the presented method, we conducted two experiments.
In the first experiment, we checked whether our video-based V t estimates match theoretical expectations for spheres (Appendix S2).
Specifically, we took 15 replicate V t estimates for each of five sphere types of different diameter d and density s (Table S1). These V t estimates closely match theoretical expectations (Figure 2a; accuracy measured as 1 − | − E| ∕ E, where is mean estimate, and E is expectation: 92.1%-100.0%). The slight underestimation of V t for spheres made of Styrofoam (30 mm) and Polyoxymethylene might be due to their somewhat rough surface and/or limited applicability of the empirical approximation for the drag coefficient (Appendix S2).
In the second experiment, we compared our video-based V t estimates to independent measurements for randomly sampled seeds of Agrimonia eupatoria and Rhinanthus minor (visually separated into "small" and "large" groups) in a vertical wind tunnel at the Institute of Agricultural Engineering in the Tropics and Subtropics, University of Hohenheim, Germany. The vertical wind tunnel accurately measures the distribution of V t for seed batches when seed falling trajectories are simple and V t is relatively high (Karaj & Müller, 2010). We placed batches of 10 seeds per species and size category in the tunnel, gradually increased wind speed (measured with a digital manometer, GDH 01 AN, Greisinger, GHM Messtechnik GmbH), and recorded the wind speed at which the first five seeds were suspended in the air, and calculated V t following Karaj and Müller (2010). These wind tunnel measurements closely matched our video-based estimates of V t (Figure 2b; accuracy: 91.1%-97.1%).
In a third experiment, we showed that V t estimates are independent of release height (length of the release tube) (Appendix S2).

| Intraspecific variability and repeatability of V t estimates
To determine the intraspecific variability and repeatability of V t estimates, 10 seeds each of A. eupatoria, Silene vulgaris, Iris pseudacorus, R. minor, and Taraxacum officinale were measured four times. To quantify intraspecific variability, we used a linear mixed-effects model with log-transformed V t as the response variable and seed identity nested within species as the random effect. Variation in V t was decomposed across the seed and species levels following Messier et al. (2010). To quantify repeatability, we used the intraclass correlation coefficient (ICC) following Wolak et al. (2012).
Across all five study species, species identity explained 98.4% of the variance in log-transformed terminal velocity, seed identity within species explained 1.0%, and the unexplained variance amounted to 0.6%. The ICC across all seeds of the study species was 0.993 (Table 1), indicating very high repeatability of V t estimates at this level.
Within individual species, seed identity explained between 39% and 91% (median 63%) of the variance in log-transformed terminal velocity, with the remaining proportion arising from the variance between repeated measures of the same seed (  Figure S7).

| Comparison to existing V t measurements
We compared our V t estimates for five plant species that cover a variety of seed sizes and shapes to the mean V t in the TRY database, which predominantly contains measurements of average velocity after seeds had fallen a certain distance and thus does not account for seed acceleration (Kattge et al., 2020). For four out of these five species, the TRY values are substantially lower than our V t estimates ( Figure 3). As expected, the underestimation of V t in the TRY database is more pronounced for fast-falling seeds that take longer to approach V t ( Figure S3).

| DISCUSS ION
The presented method for quantifying seed falling trajectories and estimating V t has six main advantages over previous methods. First, the method can quantify seed acceleration. Secondly, the method can estimate the falling trajectory ( Figure S2) and V t of single seeds in a nondestructive manner. Thirdly, our method improves on existing video-based methods that can produce erroneous estimates of V t by not accounting for the distance between the camera and the falling seed (Wyse et al., 2019). To accurately determine the vertical position of a falling seed, one needs to film the seed from two perspectives simultaneously. This could be achieved by using two cameras, but this is costly and raises the challenge of accurately synchronizing the cameras. Fourthly, the simple physical model accurately estimated V t of seeds that were still accelerating considerably ( Figure S5). This can greatly increase time efficiency because one does not have to ensure that a falling seed is close to V t in the falling corridor (Sullivan et al., 2018). It also substantially increases practicability: large A. eupatoria seeds (mean V t 6.4 m/s) only reach 99% of V t after a falling distance of 8.2 m ( Figure S3). This clearly exceeds the dimensions of most ecological laboratories. Fifthly, the evaluation functions enable users to assess the validity of the model used to estimate V t from falling trajectories. To our knowledge, no other method permits assessing the validity of V t estimates. Lastly, the method is highly automated and can rapidly estimate the V t of large numbers of seeds. While existing methods should yield reliable estimates of V t for many slow-falling seeds, the new method presented here has a greater scope of application and provides additional insights into the mechanisms of wind-driven seed dispersal.
We discuss these aspects in the following.

| Applicability of the method to different seed sizes
We successfully estimated V t of seeds ranging from 0.7 mm to 3 cm in diameter. Yet our apparatus cannot be used for very large and very small seeds. However, the method is in principle applicable to seeds of any size. Larger seeds can be accommodated by increasing the dimensions of the apparatus, whereas smaller seeds can be estimated by reducing these dimensions and/or using a higherresolution camera.

| Improving the quantification of seed dispersal
The presented method has important implications for the quantification of wind-driven seed dispersal. The first implication is that many V t estimates in existing databases (e.g. TRY; Kattge et al., 2020) probably underestimate the true V t of fast-falling seeds (Figure 3).
Such underestimation results if V t is taken to be the average falling velocity after a limited acceleration distance that is insufficient for the seed to approach V t (Figure 2). This underestimation of V t should cause mechanistic dispersal models to overestimate distances of wind-driven seed dispersal. It could be argued that this overestimation is ecologically irrelevant because fast-falling seeds invariably have short dispersal distances. However, the mechanistic model of TA B L E 1 Repeatability of V t estimates for individual seeds, estimated as the intraclass correlation coefficient (ICC) following Wolak et al. (2012).

Iris pseudacorus .372
Rhinanthus minor .208 Taraxacum officinale .887 Nathan et al. (2002) predicts that strong uplifts can even transport hickory nuts (Carya glabra), which have a very high V t (7.84 m/s), over several hundred meters (Higgins et al., 2003). Since the uplift probability depends on V t , reliable mechanistic predictions of dispersal distance require accurate V t estimates even for fast-falling seeds.
A second implication of underestimating V t is an underestimation of seed inertia, as characterized by the Lagrangian relaxation timescale τ (Equation S6). Seeds with high τ take longer to accelerate in response to gravity or changes in vertical and horizontal wind speed (Nathan et al., 2011). According to the simple physical model fitted to our data for A. eupatoria, an average seed that drops in still air from the species' average seed release height (0.45 m; Kleyer et al., 2008) has reached only 44% of V t when it hits the ground. Mechanistic models that ignore seed inertia and assume that seeds fall at V t right after seed release will thus underestimate dispersal ability, in particular for smaller plants with fast-falling seeds. However, seed inertia is not represented by most mechanistic models for wind dispersal (Nathan et al., 2011) including the widely used WALD model (Katul et al., 2005; for exceptions see Bohrer et al., 2008;Kuparinen et al., 2007). Our method provides data on seed falling trajectories that can be used to validate, inversely parameterize, and/or select between models of seed acceleration. Candidate models are the simple physical model used here or more complex models that represent effects of nonspherical seed morphologies, nonquadratic drag, seed rotation, and others (Hirata et al., 2011;Lentink et al., 2009;Schwendemann et al., 2007). Suitable acceleration models can then be incorporated into improved mechanistic models for wind-driven seed dispersal.
A third implication of the presented method is that it should advance quantification of the causes and consequences of intraspecific variation in seed dispersal (Albert et al., 2011;Chen & Giladi, 2020;Snell et al., 2019;Zhu et al., 2019). In species with low seed-level repeatability of falling trajectories and V t (such as R. minor), the videobased method can be used to quantify to what extent within-seed variability arises from measurement error versus intrinsic variability in the flight behavior of a given seed ( Figure S3). This decomposition of variability in seed dispersal has important consequences for understanding the evolution of seed dispersal: the larger the intrinsic component, the less dispersal is controlled by seed traits and hence the genotype (Schurr et al., 2009). In species with high seed-level repeatability of falling trajectories and V t (such as T. officinale), the presented method should enable the efficient dispersal phenotyping of seeds from multiple plant genotypes grown in multiple environments. This would be a crucial step toward unraveling the genetic architecture of how dispersal responds to environmental variation and quantifying how rapidly dispersal can evolve in changing environments (Ronce, 2007).

| CON CLUS IONS
We present a novel video-based method that determines the threedimensional trajectory of falling seeds and analyzes these trajectories with a simple physical model of free fall with air resistance to estimate V t . This accurate, efficient, and affordable method improves the quantification of intra-and interspecific variation in seed disper- TA B L E 2 Intraspecific variability of the measured seed terminal velocity (V t ) within individual species. Each of the 10 seeds of each species was measured four times. Intraspecific variability was quantified as the coefficient of determination (R 2 ) of the linear model in which log-transformed V t was the response variable, and seed identity was the explanatory variable.

ACK N OWLED G M ENTS
The study was supported by the German Research Foundation (DFG) (SCHU 2259/3-3 and SCHU 2259/5-2) and the Sino-German (CSC-DAAD) Postdoc Scholarship Program (57165010, 2015). We thank Prof. Angela Moles and three anonymous reviewers for the constructive comments on the early versions of the manuscript.

CO N FLI C T O F I NTE R E S T
The authors have no conflict of interest.